Characterization of Riesz spaces with topologically full center

Let $E$ be a Riesz space and let $E^{\sim}$ denote its order dual. The orthomorphisms $Orth(E)$ on $E,$ and the ideal center $Z(E)$ of $E,$ are naturally embedded in $Orth(E^{\sim})$ and $Z(E^{\sim})$ respectively. We construct two unital algebra and order continuous Riesz homomorphisms $$\gamma:((Orth(E))^{\sim})_{n}^{\sim}\rightarrow Orth(E^{\sim})\text{ }$$ and $$m:Z(E)^{\prime\prime}\rightarrow Z(E^{\sim})$$ that extend the above mentioned natural inclusions respectively. Then, the range of $\gamma$ is an order ideal in $Orth(E^{\sim})$ if and only if $m$ is surjective. Furthermore, $m$ is surjective if and only if $E$ has a topologically full center. (That is, the $\sigma(E,E^{\sim})$-closure of $Z(E)x$ contains the order ideal generated by $x$ for each $x\in E_{+}.$) As a consequence, $E$ has a topologically full center $Z(E)$ if and only if $Z(E^{\sim})=\pi\cdot Z(E)^{\prime\prime}$ for some idempotent $\pi\in Z(E)^{\prime\prime}.

On invariant subspaces of collectively compact sets of linear operators

Bir Banach uzayı üzerinde tanımlı sınırlı doğrusal operatörünün aşikar olmayan (yani ve ten farklı) kapalı (hiper)değişmez altuzaya sahip olup olmaması, ‘değişmez altuzay problemi’ olarak bilinir. Burada in bir altuzayının operatörü altında değişmez kalması, hiperdeğişmez kalması ise ile değişmeli her operatör altında değişmez kalmasıdır. Tek bir operatörü yerine bu operatörlerin bir ailesi göz önüne alındığında, nin altında (hiper)değişmez kalması, ailesine ait her operatör altında (hiper)değişmez olmasıdır. in kapalı birim yuvarını göstermek üzere, kümesi önkompakt ise ailesine birlikte kompakt denir. Bu çalışmada doğrusal operatörlerin birlikte kompakt ailelerinin (hiper)değişmez altuzayları araştırılmaktadır. Kompakt operatörlerin önkompakt ailesi birlikte kompaktır ancak bunun tersi her zaman doğru değildir. Kompakt operatörlerin önkompakt aileleri için bilinen bazı değişmez altuzay sonuçları, birlikte kompakt operatör ailelerine genişletilmektedir. Bunu yaparken, Rota-Strang spektral yarıçapı, Berger-Wang spektral yarıçapı ve deki sıfırdan farklı bir elemanı için yerel spektral yarıçapı kullanılmaktadır. Ayrıca  daki birlikte kompakt ailesinin, Berger-Wang formülünü sağladığı gösterilmektedir; burada, X in altuzaylarının tam zincirini ve, daki tüm altuzayları değişmez bırakan operatörlerin kümesini göstermektedir. Anahtar kelimeler: Değişmez altuzay, birlikte kompakt kümeler, ortak spektral yarıçap.Given a Banach space and a bounded linear operator, may or may not have a closed subspace, other than and, which is left invariant under, that is,. This work is concerned with this problem which is commonly known as the invariant subspace problem. However, instead of taking a single operator, we consider a family of linear bounded operators on infinite dimensional Banach spaces which are tied together with a strong compactness condition, known as collective compactness and we look for a common invariant subspace of elements of. A family of operators is called collectively compact if the closure of the closed unit ball of is compact under the action of. That is, is compact in. Using well-known techniques, we generalize invariant subspace results which are proven for precompact families of compact operators to collectively compact families of operators. In doing so, we use joint spectral radius, Berger-Wang spectral radius of and the local joint spectral radius for a non-zero in. A common technique to show that a multiplicative semigroup generated by has a common invariant subspace is to show that has a non-zero semigroup ideal which has a non-trivial closed invariant subspace. Employing this technique and introducing a semigroup ideal in the multiplicative semigroup, we show that if is collectively compact and then has an invariant subspace. Another results in this direction are the ones which yield a common invariant subspace for a collectively compact family of operators if for some non-zero in. If, on the other hand, is collectively compact and, then we show that has a common invariant subspace. Another case where collectively compact family has a common invariant subspace is when and is not bounded. In the final part of the work, we consider a complete chain of closed subspaces of and show that if is a collectively compact family in then we have. Here, denotes the set of operators that leave all the subspaces in invariant and denotes the set of all operators satisfying for any gap in and all. As a result of this, we relate the joint spectral radius to Ringrose?s diagonal numbers for triangularizable collectively compact sets of operators: The arbitrary complete chains of invariant subspaces for collectively compact sets of operators are then considered, and the joint spectral radius is compared with joint spectral radii of sets induced by in the quotient spaces corresponding to the gaps of the chain. So we first show that that if is a collectively family in and, then there exists only a finite number of gaps of the chain  such that. And by using this result we obtain that if is a collectively family in, then we have. We finally show that the Berger-Wang formula holds for a collectively compact family in where is a complete chain of subspaces.  Keywords: Invariant subspace, collectively compact set, joint spectral radius

Cahit Hoca için

Taha Toros Arşivi, Dosya No: 357-Cahit Arf. Not: Cahit Arf'ın anısına Bilim ve Teknik Aylık Popüler Bilim Dergisi'nin 363. sayısının ekidir.Unutma İstanbul projesi İstanbul Kalkınma Ajansı'nın 2016 yılı "Yenilikçi ve Yaratıcı İstanbul Mali Destek Programı" kapsamında desteklenmiştir. Proje No: TR10/16/YNY/010

Fonksiyon Analize Giriş

Bu bir lisans dersidir. Bu derste normlu uzaylar, tamlık, fonksiyoneller, Hahn- Banach teoremi, dualite, dönüşümler; Lebegue ölçümü, ölçülebilir fonksiyonlar, integrallenebilirlik, Lp uzaylarının tamlığı; Hilbert uzayları; kompakt, Hilbert-Schmidt ve iz sınıfından dönüşümler ile spektral teorem işlenecektir.Bu ders TÜBA Açıkders Malzemeleri Projesi kapsamında hazırlanmıştır ve TÜBA Ulusal Açıkders Malzemeleri Portalinde(http://www.acikders.org.tr) de yayınlanmaktadır

The role of bio-detection dogs in the prevention and diagnosis of infectious diseases: A systematic review

nfectious diseases have been lately considered as one of the most important global risks, which negatively impact not only the health but also the socioeconomic conditions of countries. Globalization influences the spread of infectious diseases as a result of increased travelling and interaction in humans. Thus, it is highly important to prevent and diagnose new infectious diseases by using accurate and quick diagnostic methods. Bio-detection dogs have a great potential to accurately diagnose infectious disease as they have a great ability to sense diseasespecific volatile organic compounds (VOCs) originate from infectious agents and/or pathophysiological processes in the human body. The use of these dogs to detect infectious diseases has come to focus in particular after the recent global health crisis due to the SARSCoV-2 infection.This review discusses the potential use of bio-detection dogs in the prevention and diagnosing of infectious diseases. Moreover, factors affecting the scent of the disease, e.g. VOCs, are tried to be highlighted.Bulaşıcı hastalıklar, son zamanlarda sadece ülkelerin sağlığını değil, ekonomisini de olumsuz yönde etkBulaşıcı hastalıklar, son zamanlarda sadece ülkelerin sağlığını değil, ekonomisini de olumsuz yönde etkileyen en önemli küresel risklerden biri olarak kabul edilmektedir. Küreselleşme, yeni insan seyahat modellerinin ve artan insan etkileşiminin bir sonucu olarak bulaşıcı hastalıkların yayılmasını etkilemektedir. Dolayısıyla yeni bulaşıcı hastalıkların doğru ve hızlı teşhis yöntemleri kullanılarak önlenmesi ve teşhis edilmesi büyük önem taşımaktadırileyen en önemli küresel risklerden biri olarak kabul edilmektedir. Küreselleşme, yeni insan seyahat modellerinin ve artan insan etkileşiminin bir sonucu olarak bulaşıcı hastalıkların yayılmasını etkilemektedir. Dolayısıyla yeni bulaşıcı hastalıkların doğru ve hızlı teşhis yöntemleri kullanılarak önlenmesi ve teşhis edilmesi büyük önem taşımaktadır. Biyodedektör köpekler, insan vücudundaki (pato)fizyolojik süreçler sırasında oluşan hastalığa özgü uçucu organik bileşikleri (VOC'ler) ayırt etme konusunda büyük bir yeteneğe sahip oldukları için bulaşıcı hastalıkları doğru bir şekilde teşhis etmede önemli bir potansiyele sahiptir. Bu köpeklerin enfeksiyöz hastalıkları teşhis etmek için kullanılması özellikle SARS-CoV-2 enfeksiyonu nedeniyle yakın zamanda yaşanan küresel sağlık krizinden sonra araştırmaların odak noktası haline gelmiştir. Bu derleme, bulaşıcı hastalıkların önlenmesi ve teşhisinde biyodedektör köpeklerinin potansiyel kullanımını tartışmaktadır. Ayrıca hastalık kokusunu etkileyen faktörler yani VOC'ler de aydınlatılmaya çalışılmıştır.WOS:0006362251000122-s2.0-8510381994

Mortality analysis of COVID-19 infection in chronic kidney disease, haemodialysis and renal transplant patients compared with patients without kidney disease: a nationwide analysis from Turkey

Background. Chronic kidney disease (CKD) and immunosuppression, such as in renal transplantation (RT), stand as one of the established potential risk factors for severe coronavirus disease 2019 (COVID-19). Case morbidity and mortality rates for any type of infection have always been much higher in CKD, haemodialysis (HD) and RT patients than in the general population. A large study comparing COVID-19 outcome in moderate to advanced CKD (Stages 3-5), HD and RT patients with a control group of patients is still lacking. Methods. We conducted a multicentre, retrospective, observational study, involving hospitalized adult patients with COVID-19 from 47 centres in Turkey. Patients with CKD Stages 3-5, chronic HD and RT were compared with patients who had COVID-19 but no kidney disease. Demographics, comorbidities, medications, laboratory tests, COVID-19 treatments and outcome [in-hospital mortality and combined in-hospital outcome mortality or admission to the intensive care unit (ICU)] were compared. Results. A total of 1210 patients were included [median age, 61 (quartile 1-quartile 3 48-71) years, female 551 (45.5%)] composed of four groups: Control (n = 450), HD (n = 390), RT (n = 81) and CKD (n = 289). The ICU admission rate was 266/ 1210 (22.0%). A total of 172/1210 (14.2%) patients died. The ICU admission and in-hospital mortality rates in the CKD group [114/289 (39.4%); 95% confidence interval (CI) 33.9-45.2; and 82/289 (28.4%); 95% CI 23.9-34.5)] were significantly higher than the other groups: HD = 99/390 (25.4%; 95% CI 21.3-29.9; P<0.001) and 63/390 (16.2%; 95% CI 13.0-20.4; P<0.001); RT = 17/81 (21.0%; 95% CI 13.2-30.8; P = 0.002) and 9/81 (11.1%; 95% CI 5.7-19.5; P = 0.001); and control = 36/450 (8.0%; 95% CI 5.8-10.8; P<0.001) and 18/450 (4%; 95% CI 2.5-6.2; P<0.001). Adjusted mortality and adjusted combined outcomes in CKD group and HD groups were significantly higher than the control group [hazard ratio (HR) (95% CI) CKD: 2.88 (1.52- 5.44); P = 0.001; 2.44 (1.35-4.40); P = 0.003; HD: 2.32 (1.21- 4.46); P = 0.011; 2.25 (1.23-4.12); P = 0.008), respectively], but these were not significantly different in the RT from in the control group [HR (95% CI) 1.89 (0.76-4.72); P = 0.169; 1.87 (0.81-4.28); P = 0.138, respectively]. Conclusions. Hospitalized COVID-19 patients with CKDs, including Stages 3-5 CKD, HD and RT, have significantly higher mortality than patients without kidney disease. Stages 3-5 CKD patients have an in-hospital mortality rate as much as HD patients, which may be in part because of similar age and comorbidity burden. We were unable to assess if RT patients were or were not at increased risk for in-hospital mortality because of the relatively small sample size of the RT patients in this study